Octave 3.8, jcobi/2

Average Error: 24.7 → 1.2

Time: 23.6s

Precision: binary64

Cost: 28868

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.9995)
     (/
      (+
       (* -2.0 (* (/ beta alpha) (/ beta alpha)))
       (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha))
      2.0)
     (/
      (log
       (exp
        (+
         (*
          (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
          (/ (- beta alpha) (fma 2.0 i (+ alpha beta))))
         1.0)))
      2.0))))

C

double code(double alpha, double beta, double i) {
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9995) {
		tmp = ((-2.0 * ((beta / alpha) * (beta / alpha))) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	} else {
		tmp = log(exp(((((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / fma(2.0, i, (alpha + beta)))) + 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.9995)
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(beta / alpha) * Float64(beta / alpha))) + Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha)) / 2.0);
	else
		tmp = Float64(log(exp(Float64(Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta)))) + 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(-2.0 * N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Log[N[Exp[N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99950000000000006

   1. Initial program 61.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Simplified 54.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}} \]
      Proof
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) beta)))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) (+.f64 beta 2))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 i) beta) 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha (+.f64 beta (*.f64 2 i))) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 51 points increase in error, 22 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (Rewrite<= metadata-eval (neg.f64 -1))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) -1)) 2): 8 points increase in error, 14 points decrease in error
      (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta))) -1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> fma-neg_binary64 (fma.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta) (neg.f64 -1))) 2): 14 points increase in error, 8 points decrease in error
      (/.f64 (fma.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta) (Rewrite=> metadata-eval 1)) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta)) 1)) 2): 8 points increase in error, 14 points decrease in error
      (/.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 82 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 3 points increase in error, 2 points decrease in error

   3. Taylor expanded in alpha around inf 13.2

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta \cdot \left(\beta - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right)}{{\alpha}^{2}} + \left(-1 \cdot \frac{\beta}{\alpha} + \left(\frac{\beta}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right) + -1 \cdot \left(\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right)\right)\right) - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}}{2} \]

   4. Taylor expanded in beta around inf 8.7

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   5. Simplified 5.2

      \[\leadsto \frac{\color{blue}{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      Proof
      (*.f64 -2 (*.f64 (/.f64 beta alpha) (/.f64 beta alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 -2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 beta beta) (*.f64 alpha alpha)))): 68 points increase in error, 35 points decrease in error
      (*.f64 -2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (*.f64 alpha alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 -2 (/.f64 (pow.f64 beta 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error

   if -0.99950000000000006 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 13.4

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Applied egg-rr 0.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	1.6
Cost	3140

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.9995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	7.1
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 4.165488879421093 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.6794902196593948 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.492698181997146 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.7
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 4.165488879421093 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.6794902196593948 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.492698181997146 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.0553185487033835 \cdot 10^{+164}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;i \leq 1.1495975426981173 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.8437441269717573 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{elif}\;i \leq 1.4896057041156688 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Error	10.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.0553185487033835 \cdot 10^{+164}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	14.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.0553185487033835 \cdot 10^{+164}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	17.9
Cost	460

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.358071764949556 \cdot 10^{+33}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 3.5461441633782197 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 2.175033216420716 \cdot 10^{+91}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	42.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022298 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))